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An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. (English) Zbl 0878.65091
The point of this paper is that it is worthwhile to think about finite element and finite volume methods in terms of the algebraic topology of the division of the domain into cells. The primary advantages are that vector and tensor quantities are handled consistently and that conservation laws are automatically satisfied. Furthermore, the decisions to be made by the user are reduced to the choice of the basis functions and the representation of the constitutive relations. Discretization of the heat equation is worked out as an example.
MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65M06Finite difference methods (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
55-99Algebraic topology (MSC2000)
65N06Finite difference methods (BVP of PDE)
35K05Heat equation